Let \( \alpha = 3 + 4 + 8 + 9 + 13 + 14 + \ldots \) upto \( 40 \) terms. If \( (\tan \beta)^{\frac{\alpha}{1020}} \) is a root of the equation \( x^2 + x - 2 = 0 \), \( \beta \in \left(0, \frac{\pi}{2}\right) \), then \( \sin^2 \beta + 3 \cos^2 \beta \) is equal to:
